Winning in a Coin Toss Game

Solution & Explanation

The problem presented is a classic example of a stochastic process known as the Gambler's Ruin problem. In this scenario, a gambler starts with a certain amount of money and continues to gamble until they either lose all their money or reach a target amount. The key here is to determine the probability of the gambler reaching the target amount before going broke.

Problem Breakdown:

• Initial Capital: $30
• Target Capital: $100
• Bankruptcy Point: $0
• Outcome of Each Game: Win $1 for heads, lose$1 for tails.
• Probability of Heads or Tails: 0.5 each (assuming a fair coin).

Approach:

1. Understanding the Random Walk:

   • Each coin flip represents a step in a random walk.
   • The gambler's current capital can increase by $1 (for heads) or decrease by$1 (for tails).
   • The process continues until the gambler reaches $100 or$0.

2. Expected Value and Probability:

   • The expected value of the gambler's capital remains constant at $30 throughout the game. This is because the expected change in capital per flip is zero (0.5 * 1 + 0.5 * (-1) = 0).
   • The expected value at the end of the game is a weighted sum of the possible outcomes ($100 or$0) multiplied by their respective probabilities.

3. Setting Up the Equation:

   • Let r be the probability of reaching $100.

   • The probability of reaching $0 is 1 - r.

   • The expected value equation can be set up as follows:

     $E[X] = 100 \times r + 0 \times (1-r) = 100r$

     Since we know that the expected value is $30, we substitute:

     $30 = 100r$

     Solving for r gives us:

     $r = \frac{30}{100} = 0.3$

4. Conclusion:

   • The probability that the gambler will reach $100 before going broke is 0.3, or 30%.

Additional Insights:

• Markov Chain Perspective:

  • This problem can also be approached using Markov chains, where each state represents a possible amount of money the gambler can have.
  • The transition probabilities between states are determined by the outcomes of the coin toss.

• Symmetry and Fairness:

  • The fairness of the coin ensures that the expected value remains constant, highlighting the inherent symmetry in the problem.

• Practical Implications:

  • In real-world scenarios, understanding such probability models can assist in decision-making processes involving risk and reward, such as financial investments or strategic games.

This explanation provides a comprehensive understanding of the problem and its solution, integrating concepts from probability theory and stochastic processes.